In this paper, we derive the following bound on the size of a k-wise L-intersecting family (resp. cross-L-intersecting families) modulo a prime number. This result may be considered as a modular version of Theorem 1.10 in [J.Q. Liu, S.G. Zhang, S.C. Li, H.H. Zhang, European J. Combin. 58 (2016) 166-180]. We also report a theorem which extends the well-known Alon-Babai-Suzuki theorem to two cross L-intersecting families. This is joint work with Professor Shuchao Li.

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